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Mean ergodic semigroups of operators

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Abstract

We present criteria for determining mean ergodicity of C 0-semigroups of linear operators in a sequentially complete, locally convex Hausdorff space X. A characterization of reflexivity of certain spaces X with a basis via mean ergodicity of equicontinuous C 0-semigroups acting in X is also presented. Special results become available in Grothendieck spaces with the Dunford–Pettis property.

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References

  1. Albanese A.A., Bonet J., Ricker W.J.: Mean ergodic operators in Fréchet spaces. Ann. Acad. Sci. Fenn. Math. 34, 401–436 (2009)

    MathSciNet  MATH  Google Scholar 

  2. Albanese A.A., Bonet J., Ricker W.J.: Grothendieck spaces with the Dunford–Pettis property. Positivity 14, 145–164 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Albanese, A.A., Bonet, J., Ricker, W.J.: On mean ergodic operators. In: Curbera, G.P. et al. (eds.) Vector Measures, Integration and Related Topics. Operator Theory: Advances and Applications, vol. 201, pp. 1–20. Birkhäuser, Basel (2010)

  4. Albanese A.A., Bonet J., Ricker W.J.: C 0-semigroups and mean ergodic operators in a class of Fréchet spaces. J. Math. Anal. Appl. 365, 142–157 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bonet J., Ricker W.J.: Schauder decompositions and the Grothendieck and the Dunford–Pettis properties in Köthe echelon spaces of infinite order. Positivity 11, 77–93 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bonet J., Ricker W.J.: Mean ergodicity of multiplication operators in weighted spaces of holomorphic functions. Arch. Math. 92, 428–437 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bonet J., de Pagter B., Ricker W.J.: Mean ergodic operators and reflexive Fréchet lattices. Proc. R. Soc. Edinb. Sect. A 141, 897–920 (2011)

    Article  MATH  Google Scholar 

  8. Dunford N., Schwartz J.T.: Linear Operators I: General Theory, 2nd edn. Wiley, New York (1964)

    Google Scholar 

  9. Eberlein W.F.: Abstract ergodic theorems and weak almost periodic functions. Trans. Am. Math. Soc. 67, 217–240 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  10. Edwards R.E.: Functional Analysis. Reinhart and Winston, New York (1965)

    MATH  Google Scholar 

  11. Engel K.-J., Nagel R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (1999)

    Google Scholar 

  12. Floret, K.: Weakly compact sets. In: LNM, vol. 801. Springer, Berlin (1980)

  13. Fonf V.P., Lin M., Wojtaszczyk P.: Ergodic characterizations of reflexivity in Banach spaces. J. Funct. Anal. 187, 146–162 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hille, E., Phillips, R.S.: Functional Analysis and Semigroups, 4th edn. American Math. Soc., Providence (1981, revised)

  15. Jarchow H.: Locally Convex Spaces. B.G. Teubner, Stuttgart (1981)

    Book  MATH  Google Scholar 

  16. Kelley J.L.: General Topology, Rev. Edn, D. van Nostrand Co., Princeton–New York (1961)

  17. Komatsu H.: Semi-groups of operators in locally convex spaces. J. Math. Soc. Japan 16, 230–262 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  18. Komura T.: Semigroups of operators in locally convex spaces. J. Funct. Anal. 2, 258–296 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  19. Köthe, G.: Topological Vector Spaces I, 2nd edn. Springer, Berlin (1983, revised)

  20. Köthe G.: Topological Vector Spaces II. Springer, Berlin (1979)

    Book  MATH  Google Scholar 

  21. Krengel U.: Ergodic Theorems. Walter de Gruyter, Berlin (1985)

    Book  MATH  Google Scholar 

  22. Lotz, H.P. (1984) Tauberian theorems for operators on L ∞ and similar spaces. In: Bierstedt, K.D., Fuchssteiner, B. (eds.) Functional Analyis: Surveys and Recent Results III. North Holland, Amsterdam, pp. 117–133

  23. Lotz H.P.: Uniform convergence of operators on L ∞ and similar spaces. Math. Z. 190, 207–220 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  24. Meise R., Vogt D.: Introduction to Functional Analysis. Clarendon Press, Oxford (1997)

    MATH  Google Scholar 

  25. Miyadera I.: Semigroups of operators in Fréchet spaces and applications to partial differential operators. Tôhoku Math. J. 11, 162–183 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  26. Mugnolo D.: A semigroup analogue of the Fonf–Lin–Wojtaszczyk characterization of reflexive Banach spaces with a basis. Studia Math. 164, 243–251 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  27. Piszczek K.: Quasi-reflexive Fréchet spaces and mean ergodicity. J. Math. Anal. Appl. 361, 224–233 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Piszczek K.: Barrelled spaces and mean ergodicity. RACSAM 104, 5–11 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Rudin W.: Functional Analysis. McGraw-Hill, New York (1973)

    MATH  Google Scholar 

  30. Sato R.: On a mean ergodic theorem. Proc. Am. Math. Soc. 83, 563–564 (1981)

    Article  MATH  Google Scholar 

  31. Schaefer H.H.: Banach Lattices and Positive Operators. Springer, Berlin (1974)

    Book  MATH  Google Scholar 

  32. Shaw S.-Y.: Ergodic projections of continuous and discrete semigroups. Proc. Am. Math. Soc. 78, 69–76 (1980)

    Article  MATH  Google Scholar 

  33. Yosida K.: Functional Analysis. Springer, Berlin (1980)

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica “E.De Giorgi”, Università del Salento, C.P. 193, 73100, Lecce, Italy

    Angela A. Albanese

  2. Instituto Universitario de Matemática Pura y Aplicada IUMPA, Universitat Politècnica de València, 46071, Valencia, Spain

    José Bonet

  3. Math.-Geogr. Fakultät, Katholische Universität Eichstätt-Ingolstadt, 85072, Eichstätt, Germany

    Werner J. Ricker

Authors
  1. Angela A. Albanese
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  2. José Bonet
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  3. Werner J. Ricker
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Correspondence to Angela A. Albanese.

Additional information

Research partially supported by MICINN and FEDER Project MTM2010-15200 and GV Project Prometeo/2008/101.

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Albanese, A.A., Bonet, J. & Ricker, W.J. Mean ergodic semigroups of operators. RACSAM 106, 299–319 (2012). https://doi.org/10.1007/s13398-011-0054-2

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